In a small saucepan over medium heat, mix soy sauce, 1/4 cup water, red wine vinegar, sugars, garlic and ginger . Stir with a whisk until sugar is dissolved.
In a small bowl, whisk together 2 tablespoons warm water and cornstarch until cornstarch is completely dissolved. Slowly whisk cornstarch mixture into the sauce and simmer until thickened. Remove from heat and set aside until ready to use.
Heat olive oil in a large skillet over medium-high heat. Add chicken to the skillet and season with salt and pepper. Cook chicken until no longer pink, turning over pieces to brown evenly.
Stir in about 2/3 of the teriyaki sauce to coat the chicken. Reserve the remaining sauce for drizzling or dipping.
Meanwhile, separate Romaine lettuce leaves; wash and pat dry.
Drain the juice from the pineapple and caramelize in a heavy-bottomed skillet (optional).
To assemble, place a heaping spoonful of teriyaki coated chicken in one of the lettuce leaves. Top with jack cheese, sliced green onions and a slice of pineapple. Garnish with toasted sesame seeds (optional).
Serve while chicken is still warm with extra sauce on the side for dipping, or drizzle on top before serving.
*To caramelize pineapple slices (optional), heat a large heavy bottomed skillet (not nonstick) over medium-high heat. Arrange slices in the skillet and cook, turning once, until both sides are browned and caramelized.